Ill-Posed Point Neuron Models.

John Wyller,Bjørn Fredrik Nielsen

doi:10.1186/s13408-016-0039-8

John Wyller, Bjørn Fredrik Nielsen

Open Access

https://doi.org/10.1186/s13408-016-0039-8

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Abstract

We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite precision arithmetic is used, then it is virtually impossible to guarantee the accurate numerical solution of such models. If a smooth firing rate function is employed, then standard ODE theory implies that point-neuron models are well posed. Nevertheless, in the steep firing rate regime, the problem may become close to ill posed, and the error amplification, in finite time, can be very large. This observation is illuminated by numerical experiments. We conclude that, if a steep firing rate function is employed, then minor round-off errors can have a devastating effect on simulations, unless proper error-control schemes are used.

Highlights

Modeling of electrical potentials has a long tradition in computational neuroscience
Note that in both cases the neuron fires, i.e. the change in the initial condition is not such that it has moved from one side of an unstable equilibrium to the other side
We have observed that models with a steep, but smooth, firing rate function can amplify errors to an extreme degree, which is typical for “almost ill-posed” problems

Summary

Introduction

Modeling of electrical potentials has a long tradition in computational neuroscience. One model with some physiological significance is the voltage-based system τ u (t) = −u(t) + ωSβ u(t) − uθ + q(t), t ∈ (0, T ], (1) u(0) = u0, (2). In the rate model (1)–(2), each component function ui(t) of u(t) represents the time dependent potential of the ith unit in a network of N units. The nonlinear function Sβ is called the firing rate function, {ωij } are the connectivities, and q(t) models the external drive. A detailed derivation of this model can be found in [1,2,3]

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